Gentle sleep training is much easier to start in your room, with a sidecar cot - the set up you have. So I wouldn't be entertaining the idea of putting baby in own room for a while yet. Unless you're looking to do more harsh sleep training, with more acceptance of crying/distress.
As to where to start - firstly it's probably with your expectations. If you're looking for gentle sleep training that you need to properly accept that this type of sleep training is a journey - so the celebrating of progress is in making small steps towards independence. Rather than an assumption that your baby will be independently settling as a result of a few weeks work from you. You get that sort of speedy outcomes with much harsher, non-gentle sleep training methods.
So after your expectations, where next to start?
Your start point is you sitting up, baby in a cradle hold and then breastfeeding/cuddling to sleep and then putting baby into a sidecar cot already asleep (I've assumed that from your OP, please correct me if wrong).
So progress would be getting baby used to going to sleep lying down on a mattress, rather than in your arms. So my start point would be to feed baby lying down. If you have a full sized cot as your sidecar cot, you ,may be able to lie your torso in the cot and feed baby to sleep in there. If not then breastfeed with baby lying on your bed.
Once asleep, master the art of "scooting" baby over into the sidecar. This is putting your arms under sleeping baby and push baby over into the cot, rather than lifting and putting down.
That's the first step on the "journey" towards independent sleep - baby going to sleep lying down on a mattress and not in your arms. The next stages is swapping breastfeeding to sleep with dummy sucking to sleep.
This might take the form of swapping the breastfeed for dummy/cuddle slightly earlier in progress to being asleep. The aim here is that you can put baby on your/sidecar mattress awake and then cuddle to sleep with a dummy.
Then move the cuddling into the sidecar cot. By this point you'll note you are still 100% needed to settle baby to sleep, even though you will be many weeks away from where you are now. So work towards being able to feed baby, then get ready for bed/nap, then put down awake with a dummy and you physically snuggle into the sidecar cot - but extract yourself once asleep and leave baby there.
Then over time, reduce how physically close you need to be to settle baby, until it might just be hand on baby's chest and dummy suck until into a deep sleep. Note you're still needed for the entire settling process for baby. You might be many months away from where you are now, but still need to stay with baby all the time until fully asleep. By taking the gentle route, you're baby in on that journey towards independent sleep, but realistic expectations are needed in terms of long term commitment to get there.
Then from here the sides can be back on the full sized cot. Then work on baby being less dependant on your physical touch to go to sleep.